Now we will introduce a variant of the concept of `compact
representation' from [1]}. One difference is that
we will use a less restrictive notion
of `fork' than `minority fork', because we want to avoid assuming
finiteness of the algebras considered unless finiteness is necessary for the
conclusions.
Another difference is that we
will consider subalgebras of products of algebras,
rather than subalgebras of powers of a single algebra.
Let \(\mathbf {A}_1,\dots ,\mathbf {A}_n\in {\mathcal {V}}\) , and let
\(B\subseteq A_1\times \dots \times A_n\) .
For \(i\in [n]\)  and \(\gamma ,\delta \in A_i\) 
we will say that
\((\gamma ,\delta )\)  is a fork in the \(i\) -th coordinate of \(B\) if
there exist \(b,\widehat{b}\in B\)  such that
\(b|_{[i-1]}=\widehat{b}|_{[i-1]}\quad \text{and}\quad b|_i=\gamma ,\ \ \widehat{b}|_i=\delta .\) 
